%******************************************************************************* %*********************************** Fourth Chapter ***************************** %******************************************************************************* \chapter{Quantum Measurement Backaction} % Title of the Fourth Chapter \ifpdf \graphicspath{{Chapter4/Figs/Raster/}{Chapter4/Figs/PDF/}{Chapter4/Figs/}} \else \graphicspath{{Chapter4/Figs/Vector/}{Chapter4/Figs/}} \fi %********************************** %First Section ************************************** \section{Introduction} This thesis is entirely concerned with the question of measuring a quantum many-body system using quantized light. However, so far we have only looked at expectation values in a nondestructive context where we neglect the effect of the quantum wavefunction collapse. We have shown that light provides information about various statistical quantities of the quantum states of the atoms such as their correlation functions. In general, any quantum measurement affects the system even if it doesn't physically destroy it. In our model both optical and matter fields are quantized and their interaction leads to entanglement between the two subsystems. When a photon is detected and the electromagnetic wavefunction of the optical field collapses, the matter state is also affected due to this entanglement resulting in quantum measurement backaction. Therefore, in order to determine these quantities multiple measurements have to be performed to establish a precise measurement of the expectation value which will require repeated preparations of the initial state. In the following chapters, we consider a different approach to quantum measurement in open systems and instead of considering expectation values we look at a single experimental run and the resulting dynamics due to measurement backaction. Previously, we were mostly interested in extracting information about the quantum state of the atoms from the scattered light. The flexibility in the measurement model was used to enable probing of as many different quantum properties of the ultracold gas as possible. By focusing on measurement backaction we instead investigate the effect of photodetections on the dynamics of the many-body gas as well as the possible quantum states that we can prepare instead of what information can be extracted. In this chapter, we introduce the necessary theory of quantum measurement and backaction in open systems in order to lay a foundation for the material that follows in which we apply these concepts to a bosonic quantum gas. We first introduce the concept of quantum trajectories which represent a single continuous series of photon detections. We also present an alternative approach to open systems in which the measurement outcomes are discarded. This will be useful when trying to learn about dynamical features common to every trajectory. In this case we use the density matrix formalism which obeys the master equation. This approach is more common in dissipative systems and we will highlight the differences between these two different types of open systems. We conclude this chapter with a new concept that will be central to all subsequent discussions. In our model measurement is global, it couples to operators that correspond to global properties of the quantum gas rather than single-site quantities. This enables the possibility of performing measurements that cannot distinguish certain sites from each other. Due to a lack of ``which-way'' information this leads to the creation of spatially nontrivial virtual lattices on top of the physical lattice. This turns out to have significant consequences on the dynamics of the system. \section{Quantum Trajectories} A simple intuitive concept of a quantum trajectory is that it is the path taken by a quantum state over time during a single experimental realisation. In particular, we consider states conditioned upon measurement results such as the photodetection times. Such a trajectory is generally stochastic in nature as light scattering is not a deterministic process. Furthermore, they are in general discotinuous as each detection event brings about a drastic change in the quantum state due to the wavefunction collapse of the light field. Before we discuss specifics relevant to our model of quantized light interacting with a quantum gas we present a more general overview which will be useful as some of the results in the following chapters are more general. Measurement always consists of at least two competing processes, two possible outcomes. If there is no competition and only one outcome is possible then our probe is meaningless as it does not reveal any information about the system. In its simplest form measurement consists of a series of detection events, such as the detections of photons. Even though, on an intuitive level it seems that we have defined only a single outcome, the detection event, this arrangement actually consists of two mutually exclusive outcomes. At any point in time an event either happens or it does not, a photon is either detected or the detector remains silent, also known as a null result. Both outcomes reveal some information about the system we are investigating. For example, let us consider measuring the number of atoms by measuring the number of photons they scatter. Each atom will on average scatter a certain number of photons contributing to the detection rate we observe. Therefore, if we record multiple photons at a high rate of arrival we learn that the illuminated region must contain many atoms. On the other hand, if there are few atoms to scatter the light we will observe few detection events which we interpret as a continuous series of non-detection events interspersed with the occasional detector click. This trajectory informs us that there are much fewer atoms being illuminated than previously. Basic quantum mechanics tells us that such measurements will in general affect the quantum state in some way. Each event will cause a discontinuous quantum jump in the wavefunction of the system and it will have a jump operator, $\c$, associated with it. The effect of a detection event on the quantum state is simply the result of applying this jump operator to the wavefunction, $| \psi (t) \rangle$, \begin{equation} \label{eq:jump} | \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle} {\sqrt{\langle \cd \c \rangle (t)}}, \end{equation} where the denominator is simply a normalising factor \cite{MeasurementControl}. The exact form of the jump operator $\c$ will depend on the nature of the measurement we are considering. For example, if we consider measuring the photons escaping from a leaky cavity then $\c = \sqrt{2 \kappa} \hat{a}$, where $\kappa$ is the cavity decay rate and $\hat{a}$ is the annihilation operator of a photon in the cavity field. It is interesting to note that due to renormalisation the effect of a single quantum jump is independent of the magnitude of the operator $\c$ itself. However, larger operators lead to more frequent events and thus more frequent applications of the jump operator. The null measurement outcome will have an opposing effect to the quantum jump, but it has to be treated differently as it does not occur at discrete time points like the detection events themselves \cite{MeasurementControl}. Its effect is accounted for by a modification to the isolated Hamiltonian, $\hat{H}_0$, time evolution in the form \begin{equation} | \psi (t + \mathrm{d}t) \rangle = \left\{ \hat{1} - \mathrm{d}t \left[ i \hat{H}_0 + \frac{\cd \c}{2} - \frac{\langle \cd \c \rangle (t)}{2} \right] \right\} | \psi (t) \rangle. \end{equation} The effect of both outcomes can be included in a single stochastic Schr\"{o}dinger equation given by \begin{equation} \label{eq:SSE} \mathrm{d} | \psi(t) \rangle = \left[ \mathrm{d} N(t) \left( \frac{\c} {\sqrt{ \langle \cd \c \rangle (t)}} - \hat{1} \right) + \mathrm{d} t \left( \frac{\langle \cd \c \rangle (t)}{2} - \frac{ \cd \c}{2} - i \hat{H}_0 \right) \right] | \psi(t) \rangle, \end{equation} where $\mathrm{d}N(t)$ is the stochastic increment to the number of photodetections up to time $t$ which is equal to $1$ whenever a quantum jump occurs and $0$ otherwise \cite{MeasurementControl}. Note that this equation has a straightforward generalisation to multiple jump operators, but we do not consider this possibility here at all. All trajectories that we calculate in the follwing chapters are described by the stochastic Schr\"{o}dinger equation in Eq. \eqref{eq:SSE}. The most straightforward way to solve it is to replace the differentials by small time-steps $\delta t$. Then we generate a random number $R(t)$ at every time-step and a jump is applied, i.e.~$\mathrm{d}N(t) = 1$, if \begin{equation} R(t) < \langle \cd \c \rangle (t) \delta t. \end{equation} In practice, this is not the most efficient method for simulation \cite{MeasurementControl}. Instead we will use the following method. At an initial time $t = t_0$ a random number $R$ is generated. We then propagate the unnormalised wavefunction $| \tilde{\psi} (t) \rangle$ using the non-Hermitian evolution given by \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} | \tilde{\psi} (t) \rangle = -i \left( \hat{H}_0 - i \frac{\cd \c}{2} \right) | \tilde{\psi} (t) \rangle \end{equation} up to a time $T$ such that $\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$. This problem can be solved efficienlty using standard numerical techniques. At time $T$ a quantum jump is applied according to Eq. \eqref{eq:jump} which renormalises the wavefunction as well and the process is repeated as long as desired. This formulation also has the advantage that it provides a more intuitive picture of what happens during a single trajectory. The quantum jumps are self-explanatory, but now we have a clearer picture of the effect of null outcomes on the quantum state. Its effect is entirely encoded in the non-Hermitian modification to the original Hamiltonian given by $\hat{H} = \hat{H}_0 - i \cd \c / 2$. It is now easy to see that for a jump operator, $\c$, with a large magnitude the no-event outcomes will have a more significant effect on the quantum state. At the same time they will lead to more frequent quantum jumps which have an opposing effect to the non-Hermitian evolution, because the jump condition is satisfied $\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$ more frequently. In general, the competition between these two processes is balanced and without any further external influence the distribution of outcomes over many trajectories will be entirely determined by the initial state even though each individual trajectory will be unique and conditioned on the exact detection times that occured during the given experimental run. However, individual trajectories can have features that are not present after averaging and this is why we focus our attention on single experimental runs rather than average behaviour. \begin{figure}[htbp!] \centering \includegraphics[width=1.0\textwidth]{setup} \caption[Experimental Setup with Cavity]{Atoms in an optical lattice are probed by a coherent light beam (red), and the light scattered (blue) at a particular angle is enhanced and collected by a leaky cavity. The photons escaping the cavity are detected, perturbing the atomic evolution via measurement backaction.} \label{fig:cavity} \end{figure} The quantum trajectory theory can now be very straightforwardly applied to our model of ultracold bosons in an optical lattice. However, from now on we will only consider the case when the atomic system is coupled to a single mode cavity in order to enhance light scattering in one particular direction as shown in Fig. \ref{fig:cavity}. This way we have complete control over the form of the quantum jump operator, because light scattering in different directions corresponds to different measurements as we have seen in Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have to simultaneously consider all the possible directions in which light could scatter and thus include multiple jump operators reducing our ability to control the system. The model we derived in Eq. \eqref{eq:fullH} is in fact already in a form ready for quantum trajectory simulations. The phenomologically included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$ which is the jump operator we want for measurements of photons leaking from the cavity. However, we will first simplify the system by considering the regime where we can neglect the effect of the quantum potential that builds up in the cavity. Physically, this means that whilst light scatters due to its interaction with matter, the field that builds up due to the scattered photons collecting in the cavity has a negligible effect on the atomic evolution compared to its own dynamics such as inter-site tunnelling or on-site interactions. This can be achieved when the cavity-probe detuning is smaller than the cavity decay rate, $\Delta_p \ll \kappa$ \cite{caballero2015}. However, even though the cavity field has a negligible effect on the atoms, measurement backaction will not. This effect is of a different nature. It is due to the wavefunction collapse due to the destruction of photons rather than an interaction between fields. Therefore, the final form of the Hamiltonian Eq. \eqref{eq:fullH} that we will be using in the following chapters is \begin{equation} \hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F} \end{equation} \begin{equation} \hat{H}_0 = -J \sum_{\langle i, j \rangle} \bd_i b_j + \frac{U}{2} \sum_i \n_i (\n_i - 1), \end{equation} where $\hat{H}_0$ is simply the Bose-Hubbard Hamiltonian, $\gamma = \kappa |C|^2$ is a quantity that measures the strength of the measurement and we have substituted $\a_1 = C \hat{F}$. The quantum jumps are applied at times determined by the algorithm described above and the jump operator is given by \begin{equation} \label{eq:jumpop} \c = \sqrt{2 \kappa} C \hat{F}. \end{equation} Importantly, we see that measurement introduces a new energy and time scale $\gamma$ which competes with the two other standard scales responsible for the unitary dynamics of the closed system, tunnelling, $J$, and on-site interaction, $U$. If each atom scattered light inependently a different jump operator $\c_i$ would be required for each site projecting the atomic system into a state where long-range coherence is degraded. This is a typical scenario for spontaneous emission \cite{pichler2010, sarkar2014}, or for local \cite{syassen2008, kepesidis2012, vidanovic2014, bernier2014, daley2014} and fixed-range addressing \cite{ates2012, everest2014} which are typically considered in open systems. In contrast to such situations, we consider global coherent scattering with an operator $\c$ that is nonlocal. Therefore, the effect of measurement backaction is global as well and each jump affects the quantum state in a highly nonlocal way and most importantly not only will it not degrade long-range coherence, it will in fact lead to such long-range correlations itself. \mynote{introduce citations from PRX/PRA above} \section{The Master Equation} A quantum trajectory is stochastic in nature, it depends on the exact timings of the quantum jumps which are determined randomly. This makes it difficult to obtain conclusive deterministic answers about the behaviour of single trajectories. One possible approach that is very common when dealing with open systems is to look at the unconditioned state which is obtained by averaging over the random measurement results which condition the system \cite{MeasurementControl}. The unconditioned state is no longer a pure state and thus must be described by a density matrix, \begin{equation} \label{eq:rho} \hat{\rho} = \sum_i p_i | \psi_i \rangle \langle \psi_i |, \end{equation} where $p_i$ is the probability the system is in the pure state $| \psi_i \rangle$. If more than one $p_i$ value is non-zero then the state is mixed, it cannot be represented by a single pure state. The time evolution of the density operator obeys the master equation given by \begin{equation} \dot{\hat{\rho}} = -i \left[ \hat{H}_0, \hat{\rho} \right] + \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho} \cd \c \right). \end{equation} Physically, the unconditioned state, $\rho$, represents our knowledge of the quantum system if we are ignorant of the measurement outcome (or we choose to ignore it), i.e.~we do not know the timings of the detection events. We will be using the master equation and the density operator formalism in the context of measurement. However, the exact same methods are also applied to a different class of open systems, namely dissipative systems \cite{QuantumNoise}. A dissipative system is an open system that couples to an external bath in an uncontrolled way. The behaviour of such a system is similar to a system subject under observation in which we ignore all the results. One can even think of this external coupling as a measurement who's outcome record is not accessible and thus must be represented as an average over all possible trajectories. However, there is a crucial difference between measurement and dissipation. When we perform a measurement we use the master equation to describe system evolution if we ignore the measuremt outcomes, but at any time we can look at the detection times and obtain a conditioned pure state for this current experimental run. On the other hand, for a dissipative system we simply have no such record of results and thus the density matrix predicted by the master equation, which in general will be a mixed state, represents our best knowledge of the system. In order to obtain a pure state, it would be necessary to perform an actual measurement. A definite advantage of using the master equation for measurement is that it includes the effect of any possible measurement outcome. Therefore, it is useful when extracting features that are common to many trajectories, regardless of the exact timing of the events. However, in this case we do not want to impose any specific trajectory on the system as we are not interested in a specific experimental run, but we would still like to identify the set of possible outcomes and their common properties. Unfortunately, calculating the inverse of Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of a density matrix into pure states might not even be unique. However, if a measurement leads to a projection, i.e.~the final state becomes confined to some subspace of the Hilbert space, then this will be visible in the final state of the density matrix. We will show this on an example of a qubit in the quantum state \begin{equation} \label{eq:qubit0} | \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle, \end{equation} where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states of the qubit and we consider performing a measurement in the basis $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome. The quantum state will have collapsed now to the state $ | 0 \rangle$ with probability $| \alpha |^2$ and $| 1 \rangle$ with probability $| \beta |^2$. The corresponding density matrix is given by \begin{equation} \label{eq:rho1} \hat{\rho} = \left( \begin{array}{cc} | \alpha |^2 & 0 \\ 0 & |\beta|^2 \end{array} \right), \end{equation} which is a mixed state as opposed to the initial state. We note that there are no off-diagonal terms as the system is not in a superposition between the two basis states. Therefore, the diagonal terms represent classical probabilities of the system being in either of the basis states. This is in contrast to their original interpretation when the state was given by Eq. \eqref{eq:qubit0} when they could not be interpreted as in such a way. The initial state was in a quantum superposition and thus the state was indeterminate due to the quantum uncertainty in our knowledge of the state which would have manifested itself in the density matrix as non-zero off-diagonal terms. The significance of these values being classical probabilities is that now we know that the measurement has already happened and we know with certainty that the state must be either $| 0 \rangle$ or $| 1 \rangle$. We just don't know which one until we check the result of the measurement. We have assumed that it was a discrete wavefunction collapse that lead to the state in Eq. \eqref{eq:rho1} in which case the conclusion we reached was obvious. However, the nature of the process that takes us from the initial state to the final state with classical uncertainty does not matter. The key observation is that regardless of the trajectory taken, if the final state is given by Eq. \eqref{eq:rho1} we will definitely know that our state is either in the state $| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the two basis states. Therefore, if we obtained this density matrix as a result of applying the time evolution given by the master equation we would be able to identify the final states of individual trajectories even though we have no information about the individual trajectories themself. This is analogous to an approach in which decoherence due to coupling to the environment is used to model the wavefunction collapse \cite{zurek2002}, but here we will be looking at projective effects due to weak measurement. Here we have considered a very simple case of a Hilbert space with two non-degenerate basis states. In the following chapters we will generalise the above result to larger Hilbert spaces with multiple degenerate subspaces which are of much greater interest as they reveal nontrivial dynamics in the system. \section{Global Measurement and ``Which-Way'' Information} We have already mentioned that one of the key features of our model is the global nature of the measurement operators. A single light mode couples to multiple lattice sites from where atoms scatter the light coherently into a single mode which we enhance and collect with a cavity. If atoms at different lattice sites scatter light with a different phase or magnitude we will be able to identify which atoms contributed to the light we detected. However, if they scatter the photons in phase and with the same amplitude then we have no way of knowing which atom emitted the photon, we have no ``which-way'' information. When we were considering nondestructive measurements and looking at expectation values, this had no consequence on our results as we were simply interested in probing the quantum correlations of a given ground state and whether two sites were distinguishable or not was irrelevant. Now, on the other hand, we are interested in the effect of these measurements on the dynamics of the system. The effect of measurement backaction will depend on the information that is encoded in the detected photon. If a scattered mode cannot distinuguish between two different lattice sites then we have no information about the distribution of atoms between those two sites. Therefore, all quantum correlations between the atoms in these sites are unaffected by the backaction whilst their correlations with the rest of the system will change as the result of the quantum jumps. \begin{figure}[htbp!] \centering \includegraphics[width=1.0\textwidth]{1DModes} \caption[1D Modes due to Measurement Backaction]{The coefficients, and thus the operator $\hat{D}$ is made periodic with a period of three lattice sites. Therefore, the coefficients $J_{i,i}$ will repeat every third site making atoms in those sites indistinguishable to the measurement. Physically, this is due to the fact that this periodic arrangement causes the atoms within a single mode to scatter light with the same phase and amplitude. The scattered light contains no information which can be used to determine the atom distribution.} \label{fig:1dmodes} \end{figure} The quantum jump operator for our model is given by $\c = \sqrt{2 \kappa} C \hat{F}$ and we know from Eq. \eqref{eq:F} that we have a large amount of flexibility in tuning $\hat{F}$ via the geometry of the optical setup. A cavity aligned at a different angle will correspond to a different measurement. We will consider the case when $\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the argument depends on geometry rather than the exact nature of the operator it straightforwardly generalises to other measurement operators, including the case when $\hat{F} = \hat{B}$ where the bonds (inter-site operators) play the role of lattice sites. The operator $\hat{D}$ is given by \begin{equation} \hat{D} = \sum_i J_{i,i} \n_i, \end{equation} where the coefficients $J_{i,i}$ are determined from Eq. \eqref{eq:Jcoeff}. These coefficients represent the coupling strength between the atoms and the light modes and thus their spatial variation can be easily tuned by the geometry of the optical fields. We are in particular interested in making these coefficients degenerate across a number of lattice sites as shown in Figs. \ref{fig:1dmodes} and \ref{fig:2dmodes}. Note that they do not have to be periodic, but it is much easier to make them so. This makes all lattice sites with the same value of $J_{i,i}$ indistinguishable to the measurement thus partitioning the lattice into a number of distinct zones which we will refer to as modes. It is crucial to note that these partitions in general are not neighbours of each other, they are not localised, they overlap in a nontrivial way, and the patterns can be made more complex in higher dimensions as shown in Fig. \ref{fig:2dmodes} or with a more sophisticated optical setup \cite{caballero2016}. This has profound consquences as it can lead to the creation of long-range nonlocal correlations between lattice sites \cite{mazzucchi2016, kozlowski2016zeno}. The measurement does not know which atom within a certain mode scattered the light, there is no ``which-way'' information. Therefore, from the point of view of the observer's knowledge all atoms within a mode are identical regardless of their spatial separation. This effect can be used to create virtual lattices on top of the physical lattice. \begin{figure}[htbp!] \centering \includegraphics[width=1.0\textwidth]{2DModes} \caption[2D Modes due to Measurement Backaction]{ More complex patterns of virtual lattice can be created in higher dimensions. With a more complicated setup even more complicated geometries are possible.} \label{fig:2dmodes} \end{figure} We will now look at a few practical examples. The simplest case is to measure in the diffraction maximum such that $\a_1 = C \hat{N}_K$, where $\hat{N}_K = \sum_j^K \hat{n}_j$ is the number of atoms in the illuminated area. If the whole lattice is illuminated we effectively have a single mode as $J_{i,i} = 1$ for all sites. If only a subset of lattice sites is illuminated $K < M$ then we have two modes corresponding to the illuminated and unilluminated sites. It is actually possible to perform such a measurement in a nonlocal way by arranging every other site (e.g.~all the even sites) to be at a node of both the cavity and the probe. The resulting field measures the number of atoms in the remaining sites (all the odd sites) in a global manner. It does not know how these atoms are distributed among these sites as we do not have access to the individual sites. This two mode arrangement is shown in the top panel of Fig. \ref{fig:twomodes}. A different two-mode arrangement is possible by measuring in the diffraction minimum such that each site scatters in anti-phase with its neighbours as shown in the bottom panel of Fig. \ref{fig:twomodes}. \begin{figure}[htbp!] \centering \includegraphics[width=0.7\textwidth]{TwoModes1} \includegraphics[width=0.7\textwidth]{TwoModes2} \caption[Two Mode Partitioning]{Top: only the odd sites scatter light leading to a measurement of $\hat{N}_\mathrm{odd}$ and an effectove partitiong into even and odd sites. Bottom: this also partitions the lattice into odd and even sites, but this time atoms at all sites scatter light, but in anti-phase with their neighbours.} \label{fig:twomodes} \end{figure} This can approach can be generalised to an arbitrary number of modes, $Z$. For this we will conisder a deep lattice such that $J_{i,i} = u_1^* (\b{r}) u_0 (\b{r})$. We will take the probe beam to be incident normally at a 1D lattice so that $u_0 (\b{r}) = 1$. Therefore, the final form of the scattered light field is given by \begin{equation} \label{eq:Dmodes} \a_1 = C \hat{D} = C \sum_m^K \exp\left[-i k_1 m d \sin \theta_1 \right] \hat{n}_j. \end{equation} From this equation we see that it can be made periodic with a period $Z$ when \begin{equation} k_1 d \sin \theta_1 = 2\pi R / Z, \end{equation} where $R$ is just some integer and $R/Z$ are is a fraction in its simplest form. This partitions the 1D lattice in exactly $Z > 1$ modes by making every $Z$th lattice site scatter light with exactly the same phase. It is interesting to note that these angles correspond to the $K-1$ classical diffraction minima.